Theorem 3bior1fd 1438

Description: A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 420. (Contributed by Alexander van der Vekens, 8-Sep-2017.)

Hypothesis

Ref	Expression
3biorfd.1	|- ( ph -> -. th )

Assertion

Ref	Expression
3bior1fd	|- ( ph -> ( ( ch \/ ps ) <-> ( th \/ ch \/ ps ) ) )

Proof of Theorem 3bior1fd

Step	Hyp	Ref	Expression
1		3biorfd.1	. . 3 |- ( ph -> -. th )
2		biorf 420	. . 3 |- ( -. th -> ( ( ch \/ ps ) <-> ( th \/ ( ch \/ ps ) ) ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( ( ch \/ ps ) <-> ( th \/ ( ch \/ ps ) ) ) )
4		3orass 1040	. 2 |- ( ( th \/ ch \/ ps ) <-> ( th \/ ( ch \/ ps ) ) )
5	3, 4	syl6bbr 278	1 |- ( ph -> ( ( ch \/ ps ) <-> ( th \/ ch \/ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   \/ w3o 1036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-3or 1038

This theorem is referenced by:  3bior1fand  1439  3bior2fd  1440  nb3grprlem2  26283